Failures of GCH and the level by level equivalence between strong compactness and supercompactness

We force and obtain three models in which level by level equivalence between strong compactness and supercompactness holds and in which, below the least supercompact cardinal, GCH fails unboundedly often. In two of these models, GCH fails on a set having measure 1 with respect to certain canonical measures. There are no restrictions in all of our models on the structure of the class of supercompact cardinals. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)